Reasons why Abstract Algebra is valuable to math ed majors (and math majors)
Text: Contemporary Abstract Algebra, by Joseph A. Gallian (9th edition or later)
Authors Website
Writing proofs and presenting your assignments are essential skills in this course. I encourage all students to review the final chapter of Professor Holmes’s textbook before submitting their work.
Extra reference: Randall R. Holmes
Auburn University, Abstract Algebra textbook
Content of this course:
This course aims to introduce you to the practice of mathematics and independent thinking, and develop your ability to notice connections that reveal deeper, unseen insights. It also focuses on clearly articulating your understanding.
Teaching methodology of this course:
I run this class using Inquiry-based learning in which students are deeply engaged in a rich mathematical understanding environment, and regular opportunities are given for students to collaborate with peers and instructors with the goal of mathematical thinking, learning, and writing. Also, as an instructor, I identify students’ current knowledge, misconceptions, and reasoning strategies, enabling me to tailor instruction to better support learning. As a result, I can focus on designing the worksheets and assessments in a way that is accessible and useful for all students (equity).
I like to remind the students who take this course of the following content from Frederick Goodman, Professor Emeritus, University of Iowa
“You must have patience, or learn patience, and you must have time. You can’t learn these things without getting frustrated, and you can’t learn them in a hurry. If you can get someone else to explain how to solve the problems, you will learn something, but not patience, and not persistence, and not vision. So rely on yourself as far as possible. But rely on your teacher as well. Your teacher will give you hints, suggestions, and insights that can help you see for yourself. A book alone cannot do this, because it cannot listen to you and respond.” (Frederick Goodman, Algebra Abstract and Concrete, a note to reader page xii)
Covered Content:
Week 1: Review of basic number theory, induction, etc. Formal definition of groups and subgroups
Week 2: Cyclic groups
Week 3: Permutation groups
Week 4: Isomorphisms
Week 5: Cosets and Lagrange’s theorem
Week 6: Normal subgroups and factor groups
Week 7: Rings
Week 8: Integral domains
Week 9: Ideals
Week 10: Ring homomorphism
Week 11: Polynomial rings
Week 12: Divisibility in Integral Domains